Determining implicit equation of conic section from quadratic rational Bézier curve using Gröbner basis

The Gröbner Basis is a subset of finite generating polynomials in the ideal of the polynomial ring k[x 1,…,xn ]. The Gröbner basis has a wide range of applications in various areas of mathematics, including determining implicit polynomial equations. The quadratic rational Bézier curve is a rational parametric curve that is generated by three control points P 0(x 0,y 0), P 1(xi ,yi ), P 2(x 2,y 2) in ℝ2 and weights ω 0, ω 1, ω 2, where the weights ω i are corresponding to control points Pi (xi, yi ), for i = 0,1, 2. According to Cox et al (2007), the quadratic rational Bézier curve can represent conic sections, such as parabola, hyperbola, ellipse, and circle, by defining the weights ω 0 = ω 2 = 1 and ω 1 = ω for any control points P 0(x 0, y 0), P 1(x 1, y 1), and P 2(x 2, y 2). This research is aimed to obtain an implicit polynomial equation of the quadratic rational Bézier curve using the Gröbner basis. The polynomial coefficients of the conic section can be expressed in the term of control points P 0(x 0, y 0), P 1(x 1, y 1), P 2(x 2, y 2) and weight ω, using Wolfram Mathematica. This research also analyzes the effect of changes in weight ω on the shape of the conic section. It shows that parabola, hyperbola, and ellipse can be formed by defining ω = 1, ω > 1, and 0 < ω < 1, respectively.

Teaching the Notion Conic Section with Computer

The article discusses the problem of introducing and constructing mathematical concepts using a computer. The Wolfram Mathematica 12 symbolic calculation system is used at each stage of the complex spiral process to form the notion of conic section and the related concepts of focus, directrix and eccentricity. The nature of these notions implies the use of appropriate animations, 3D graphics and symbolic calculations. Our vision of the process of formation of mathematical concepts is presented. The notions ellipse, parabola and hyperbola are defined as the intersection of a conical surface with a plane not containing the vertex of the conical surface. The conical section is represented as a geometric location of points on the plane for which the ratio of the distance to the focus to the distance to the directrix is a constant value. The lines of hyperbola and ellipse are determined by their foci. The equivalence of different definitions for conical sections is commented.

Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry

I present a systematic interpretation of the foundational purpose of constructions in ancient Greek geometry. I argue that Greek geometers were committed to an operationalist foundational program, according to which all of mathematics—including its entire ontology and epistemology—is based entirely on concrete physical constructions. On this reading, key foundational aspects of Greek geometry are analogous to core tenets of 20th-century operationalist/positivist/constructivist/intuitionist philosophy of science and mathematics. Operationalism provides coherent answers to a range of traditional philosophical problems regarding classical mathematics, such as the epistemic warrant and generality of diagrammatic reasoning, superposition, and the relation between constructivism and proof by contradiction. Alleged logical flaws in Euclid (implicit diagrammatic reasoning, superposition) can be interpreted as sound operationalist reasoning. Operationalism also provides a compelling philosophical motivation for the otherwise inexplicable Greek obsession with cube duplication, angle trisection, and circle quadrature. Operationalism makes coherent sense of numerous specific choices made in this tradition, and suggests new interpretations of several solutions to these problems. In particular, I argue that: Archytas’s cube duplication was originally a single-motion machine; Diocles’s cissoid was originally traced by a linkage device; Greek conic section theory was thoroughly constructive, based on the conic compass; in a few cases, string-based constructions of conic sections were used instead; pointwise constructions of curves were rejected in foundational contexts by Greek mathematicians, with good reason. Operationalism enables us to view the classical geometrical tradition as a more unified and philosophically aware enterprise than has hitherto been recognised.

Vision-based Size Estimation and Centroid Positioning of Partially Occluded Fruits

The objective of this study was to propose a simple and efficient image processing algorithm for estimating the size and centroid of partially occluded round fruits. In the proposed method, the size and centroid of partially occluded fruit were estimated based on the mathematical properties of the arc-radius. The experimental tests were conducted in a laboratory with orange, Sunkist, apple, and tomato fruits by setting different occlusion conditions. The occlusion percentage was varied between 0% and 90%. The accuracy and processing time of the proposed method were compared with that of the widely-used conic-section circle fitting method. The results showed that the proposed method resulted in an overall accuracy of 95.1% and processing time of 0.66 s, as opposed to 60.2% and 0.68 s, respectively, using the conic-section equation. Compared with the conic-section equation, the proposed method was able to give a more robust prediction, even for low resolution images.

CONTEXTUALIZED LEARNING MODULES IN BRIDGING STUDENTS’ LEARNING GAPS IN CALCULUS WITH ANALYTIC GEOMETRY THROUGH INDEPENDENT LEARNING

The transition of the educational system in the Philippines vastly affects basic and higher education. A mismatch of pre-requisite Mathematics learning competencies from the basic education level occurred when the student reached higher education. This descriptive-developmental method of the study utilized the developed contextualized learning modules for the bridging course on the identified learning gaps in Calculus with Analytic Geometry for the Bachelor of Secondary Education (BSEd) major in Mathematics. Real-world concepts and situations featuring the Province of Sorsogon, Philippines were integrated into the learning modules while promoting independent learning. The content, format, presentations and organizations, accuracy, and up-to-datedness of information of the learning modules passed the evaluation of 13 experts (Mathematics Professors) from the different Higher Education Institutions (HEIs) in the Bicol Region, Philippines. Also, the 18 student participants were very much satisfied with the utilization of the learning modules that bridged their learning gaps in the conic section through independent learning.